SECOND LAW OF THERMODYNAMICS

SECOND LAW OF THERMODYNAMICS

The Second Law of Thermodynamics is used to determine the maximum efficiency of any process. A comparison can then be made between the maximum possible efficiency and the actual efficiency obtained.

EO 1.25

STATE the Second Law of Thermodynamics.

EO 1.26

Using the Second Law of Thermodynamics, DETERMINE the maximum possible efficiency of a system.

EO 1.27

Given a thermodynamic system, CONDUCT an analysis using the Second Law of Thermodynamics.

EO 1.28

Given a thermodynamic system, DESCRIBE the method used to determine:

a. The maximum efficiency of the system

b. The efficiency of the components within the system EO 1.29

DIFFERENTIATE between the path for an ideal process and that for a real process on a T-s or h-s diagram.

EO 1.30 Given a T-s or h-s diagram for a system EVALUATE:

a. System efficiencies

b. Component efficiencies

EO 1.31 DESCRIBE how individual factors affect system or component efficiency.

Second Law of Thermodynamics

One of the earliest statements of the Second Law of Thermodynamics was made by R. Clausius in 1850. He stated the following.

It is impossible to construct a device that operates in a cycle and produces no effect other than the removal of heat from a body at one temperature and the absorption of an equal quantity of heat by a body at a higher temperature.

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With the Second Law of Thermodynamics, the limitations imposed on any process can be studied to determine the maximum possible efficiencies of such a process and then a comparison can be made between the maximum possible efficiency and the actual efficiency achieved. One of the areas of application of the second law is the study of energy-conversion systems. For example, it is not possible to convert all the energy obtained from a nuclear reactor into electrical energy. There must be losses in the conversion process. The second law can be used to derive an expression for the maximum possible energy conversion efficiency taking those losses into account. Therefore, the second law denies the possibility of completely converting into work all of the heat supplied to a system operating in a cycle, no matter how perfectly designed the system may be. The concept of the second law is best stated using Max Planck’s description:

It is impossible to construct an engine that will work in a complete cycle and produce no other effect except the raising of a weight and the cooling of a heat reservoir.

The Second Law of Thermodynamics is needed because the First Law of Thermodynamics does not define the energy conversion process completely. The first law is used to relate and to evaluate the various energies involved in a process. However, no information about the direction of the process can be obtained by the application of the first law. Early in the development of the science of thermodynamics, investigators noted that while work could be converted completely into heat, the converse was never true for a cyclic process. Certain natural processes were also observed always to proceed in a certain direction (e.g., heat transfer occurs from a hot to a cold body). The second law was developed as an explanation of these natural phenomena.

Entropy

One consequence of the second law is the development of the physical property of matter termed entropy (S). Entropy was introduced to help explain the Second Law of Thermodynamics. The change in this property is used to determine the direction in which a given process will proceed. Entropy can also be explained as a measure of the unavailability of heat to perform work in a cycle. This relates to the second law since the second law predicts that not all heat provided to

a cycle can be transformed into an equal amount of work, some heat rejection must take place. The change in entropy is defined as the ratio of heat transferred during a reversible process to the absolute temperature of the system.

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(For a reversible process)

T abs where

∆ S = the change in entropy of a system during some process (Btu/°R) ∆ Q =

the amount of heat added to the system during the process (Btu) T abs =

the absolute temperature at which the heat was transferred (°R) The second law can also be expressed as ∆ S ≥ O for a closed cycle. In other words, entropy must

increase or stay the same for a cyclic system; it can never decrease. Entropy is a property of a system. It is an extensive property that, like the total internal energy

or total enthalpy, may be calculated from specific entropies based on a unit mass quantity of the system, so that S = ms. For pure substances, values of the specific entropy may be tabulated along with specific enthalpy, specific volume, and other thermodynamic properties of interest. One place to find this tabulated information is in the steam tables described in a previous chapter (refer back to Figure 19).

Specific entropy, because it is a property, is advantageously used as one of the coordinates when representing a reversible process graphically. The area under a reversible process curve on the T-s diagram represents the quantity of heat transferred during the process.

Thermodynamic problems, processes, and cycles are often investigated by substitution of reversible processes for the actual irreversible process to aid the student in a second law analysis. This substitution is especially helpful because only reversible processes can be depicted on the diagrams (h-s and T-s, for example) used for the analysis. Actual or irreversible processes cannot

be drawn since they are not a succession of equilibrium conditions. Only the initial and final conditions of irreversible processes are known; however, some thermodynamics texts represent an irreversible process by dotted lines on the diagrams.

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Carnot’s Principle

With the practice of using reversible processes, Sadi Carnot in 1824 advanced the study of the second law by disclosing a principle consisting of the following propositions.

1. No engine can be more efficient than a reversible engine operating between the same high temperature and low temperature reservoirs. Here the term heat reservoir is taken to mean either a heat source or a heat sink.

2. The efficiencies of all reversible engines operating between the same constant temperature reservoirs are the same.

3. The efficiency of a reversible engine depends only upon the temperatures of the heat source and heat receiver.

Carnot Cycle

The above principle is best demonstrated with a simple cycle (shown in Figure 21) and an example of a proposed heat power cycle. The cycle consists of the following reversible processes.

1-2: adiabatic compression from T C to T H due to work performed on fluid. 2-3: isothermal expansion as fluid expands when heat is added to the fluid at

temperature T H .

3-4: adiabatic expansion as the fluid performs work during the expansion process and

temperature drops from T H to T C .

4-1: isothermal compression as the fluid contracts when heat is removed from the fluid

at temperature T C .

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This cycle is known as a Carnot Cycle. The heat input (Q H ) in a Carnot Cycle is graphically represented on Figure 21 as the area under line 2-3. The heat rejected (Q C ) is graphically represented as the area under line 1-4. The difference between the heat added and the heat rejected is the net work (sum of all work processes), which is represented as the area of rectangle 1-2-3-4.

Figure 21 Carnot Cycle Representation

The efficiency ( η ) of the cycle is the ratio of the net work of the cycle to the heat input to the cycle. This ratio can be expressed by the following equation.

(Q H -Q C )/Q H = (T H -T C )/T H

(1-23) where:

1 - (T C /T H )

cycle efficiency

designates the low-temperature reservoir (°R)

designates the high-temperature reservoir (°R)

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Equation 1-23 shows that the maximum possible efficiency exists when T H is at its largest possible value or when T C is at its smallest value. Since all practical systems and processes are really irreversible, the above efficiency represents an upper limit of efficiency for any given system operating between the same two temperatures. The system’s maximum possible efficiency would be that of a Carnot efficiency, but because Carnot efficiencies represent reversible processes, the actual system will not reach this efficiency value. Thus, the Carnot efficiency serves as an unattainable upper limit for any real system’s efficiency. The following example demonstrates the above principles.

Example 1: Carnot Efficiency An inventor claims to have an engine that receives 100 Btu of heat and produces 25 Btu

of useful work when operating between a source at 140°F and a receiver at 0°F. Is the claim a valid claim?

Solution: T o

H = 140

F + 460 = 600°R

C =0

F + 460 = 460°R

η = (600-460)/600 x 100 = 23.3% Claimed efficiency = 25/100 = 25% Therefore, the claim is invalid.

The most important aspect of the second law for our practical purposes is the determination of maximum possible efficiencies obtained from a power system. Actual efficiencies will always

be less than this maximum. The losses (friction, for example) in the system and the fact that systems are not truly reversible preclude us from obtaining the maximum possible efficiency. An illustration of the difference that may exist between the ideal and actual efficiency is presented in Figure 22 and the following example.

Example 2: Actual vs. Ideal Efficiency The actual efficiency of a steam cycle is 18.0%. The facility operates from a steam

source at 340°F and rejects heat to atmosphere at 60°F. Compare the Carnot efficiency to the actual efficiency.

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Figure 22 Real Process Cycle Compared to Carnot Cycle

as compared to 18.0% actual efficiency. An open system analysis was performed using the First Law of Thermodynamics in the previous

chapter. The second law problems are treated in much the same manner; that is, an isolated, closed, or open system is used in the analysis depending upon the types of energy that cross the boundary. As with the first law, the open system analysis using the second law equations is the more general case, with the closed and isolated systems being "special" cases of the open system. The solution to second law problems is very similar to the approach used in the first law analysis.

Figure 23 illustrates the control volume from the viewpoint of the second law. In this diagram, the fluid moves through the control volume from section in to section out while work is delivered external to the control volume. We assume that the boundary of the control volume is at some environmental temperature and that all of the heat transfer (Q) occurs at this boundary. We have already noted that entropy is a property, so it may be transported with the flow of the fluid into and out of the control volume, just like enthalpy or internal energy. The entropy flow into the

control volume resulting from mass transport is, therefore, m ˙ in s in , and the entropy flow out of the control volume is m ˙ out s out , assuming that the properties are uniform at

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sections in and out. Entropy may also be added to the control volume because of heat transfer at the boundary of the control volume.

Figure 23 Control Volume for Second Law Analysis

A simple demonstration of the use of this form of system in second law analysis will give the student a better understanding of its use.

Example 3: Open System Second Law Steam enters the nozzle of a steam turbine with a velocity of 10 ft/sec at a pressure of

100 psia and temperature of 500°F at the nozzle discharge. The pressure and temperature are 1 atm at 300°F. What is the increase in entropy for the system if the mass flow rate is 10,000 lbm/hr?

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Solution: ms ˙ in ˙p ms ˙ out where ˙p = entropy added to the system

˙p m (s ˙ out s in ) s in

1.7088 Btu/lbm -°R (from steam tables)

s out =

1.8158 Btu/lbm°R (from steam tables)

˙p/ ˙ o m = s

s in 1.8158 1.7088 Btu/lbm- R

out

˙p/ ˙ m =

0.107 Btu/lbm -°R

1070 Btu/lbm -°R. = entropy added to the system

It should always be kept in mind that the Second Law of Thermodynamics gives an upper limit (which is never reached in physical systems) to how efficiently a thermodynamic system can perform.

A determination of that efficiency is as simple as knowing the inlet and exit temperatures of the overall system (one that works in a cycle) and applying Carnot’s efficiency equation using these temperatures in absolute degrees.

Diagrams of Ideal and Real Processes

Any ideal thermodynamic process can be drawn as a path on a property diagram, such as a T-s or an h-s diagram. A real process that approximates the ideal process can also be represented on the same diagrams (usually with the use of dashed lines).

In an ideal process involving either a reversible expansion or a reversible compression, the entropy will be constant. These isentropic processes will be represented by vertical lines on either T-s or h-s diagrams, since entropy is on the horizontal axis and its value does not change.

A real expansion or compression process operating between the same pressures as the ideal process will look much the same, but the dashed lines representing the real process will slant slightly towards the right since the entropy will increase from the start to the end of the process. Figures 24 and 25 show ideal and real expansion and compression processes on T-s and h-s diagrams.

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Figure 24 Expansion and Compression Processes Figure 25 Expansion and Compression Processes on T-s Diagram

on h-s Diagram

Power Plant Components

In order to analyze a complete power plant steam power cycle, it is first necessary to analyze the elements which make up such cycles. (See Figure 26) Although specific designs differ, there are three basic types of elements in power cycles, (1) turbines, (2) pumps and (3) heat exchangers. Associated with each of these three types of elements is a characteristic change in the properties of the working fluid.

Previously we have calculated system efficiency by knowing the temperature of the heat source and the heat sink. It is also possible to calculate the efficiencies of each individual component.

The efficiency of each type of component can be calculated by comparing the actual work produced by the component to the work that would have been produced by an ideal component operating isentropically between the same inlet and outlet conditions.

Figure 26 Steam Cycle

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A steam turbine is designed to extract energy from the working fluid (steam) and use it to do work in the form of rotating the turbine shaft. The working fluid does work as it expands through the turbine. The shaft work is then converted to electrical energy by the generator. In the application of the first law, general energy equation to a simple turbine under steady flow conditions, it is found that the decrease in the enthalpy of the working fluid H in -H out equals the work done by the working fluid in the turbine (W t ).

H in H out W t

(1-24)

(1-25) where:

m (h ˙ in h out ) w ˙ t

H in = enthalpy of the working fluid entering the turbine (Btu)

H out = enthalpy of the working fluid leaving the turbine (Btu) W t = work done by the turbine (ft-lb f )

m ˙ = mass flow rate of the working fluid (lb m /hr)

h in = specific enthalpy of the working fluid entering the turbine (Btu/lbm)

h out = specific enthalpy of the working fluid leaving the turbine (Btu/lbm) w ˙ t = power of turbine (Btu/hr)

These relationships apply when the kinetic and potential energy changes and the heat losses of the working fluid while in the turbine are negligible. For most practical applications, these are valid assumptions. However, to apply these relationships, one additional definition is necessary. The steady flow performance of a turbine is idealized by assuming that in an ideal case the working fluid does work reversibly by expanding at a constant entropy. This defines the so- called ideal turbine. In an ideal turbine, the entropy of the working fluid entering the turbine S in equals the entropy of the working fluid leaving the turbine.

S in =S out

s in =s out

where: o S

= entropy of the working fluid entering the turbine (Btu/ R) S out = entropy of the working fluid leaving the turbine (Btu/ o R) s o

in

= specific entropy of the working fluid entering the turbine (Btu/lbm - R) s o

in

out = specific entropy of the working fluid leaving the turbine (Btu/lbm - R) The reason for defining an ideal turbine is to provide a basis for analyzing the performance of

turbines. An ideal turbine performs the maximum amount of work theoretically possible.

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An actual turbine does less work because of friction losses in the blades, leakage past the blades and, to a lesser extent, mechanical friction. Turbine efficiency η t , sometimes called isentropic turbine efficiency because an ideal turbine is defined as one which operates at constant entropy,

is defined as the ratio of the actual work done by the turbine W t, actual to the work that would be done by the turbine if it were an ideal turbine W t,ideal .

W η t,actual

(1-26)

W t,ideal

(h in h out ) η actual

(1-27)

(h in h out ) ideal

where: η t

= turbine efficiency (no units)

W t,actual

= actual work done by the turbine (ft-lbf)

W t,ideal

= work done by an ideal turbine (ft-lbf)

(h in -h out ) actual = actual enthalpy change of the working fluid (Btu/lbm) (h in -h out ) ideal = actual enthalpy change of the working fluid in an ideal turbine

(Btu/lbm)

In many cases, the turbine efficiency η t has been determined

independently. This permits the actual work done to be calculated directly by multiplying the turbine efficiency η t by the work done by an ideal turbine under the same conditions. For small turbines, the turbine efficiency is generally 60% to 80%; for large turbines, it is generally about 90%.

The actual and idealized performances of a turbine may be compared conveniently using a T-s diagram. Figure 27 shows such a comparison. The ideal case is a Figure 27 Comparison of Ideal and Actual Turbine Performances constant entropy. It is represented by a vertical line on the T-s diagram. The actual turbine involves an increase in entropy. The

smaller the increase in entropy, the closer the turbine efficiency η t is to 1.0 or 100%.

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A pump is designed to move the working fluid by doing work on it. In the application of the first law general energy equation to a simple pump under steady flow conditions, it is found that the increase in the enthalpy of the working fluid H out -H in equals the work done by the pump, W p , on the working fluid.

H out H in W p

(1-28)

(1-29) where:

m (h ˙ out h in ) w ˙ p

H out

= enthalpy of the working fluid leaving the pump (Btu)

H in

= enthalpy of the working fluid entering the pump (Btu)

= work done by the pump on the working fluid (ft-lbf)

m ˙ = mass flow rate of the working fluid (lbm/hr)

h out = specific enthalpy of the working fluid leaving the pump (Btu/lbm)

h in = specific enthalpy of the working fluid entering the pump (Btu/lbm) w ˙ p

= power of pump (Btu/hr) These relationships apply when the kinetic and potential energy changes and the heat losses of

the working fluid while in the pump are negligible. For most practical applications, these are valid assumptions. It is also assumed that the working fluid is incompressible. For the ideal case, it can be shown that the work done by the pump W p is equal to the change in enthalpy across the ideal pump.

W p ideal = (H out -H in ) ideal (1-30) w ˙ p ideal = m ˙ (h out -h in ) ideal

(1-31) where:

= work done by the pump on the working fluid (ft-lbf)

H out

= enthalpy of the working fluid leaving the pump (Btu)

H in

= enthalpy of the working fluid entering the pump (Btu)

w ˙ p = power of pump (Btu/hr) m ˙

= mass flow rate of the working fluid (lbm/hr)

h out = specific enthalpy of the working fluid leaving the pump (Btu/lbm)

h in = specific enthalpy of the working fluid entering the pump (Btu/lbm) The reason for defining an ideal pump is to provide a basis for analyzing the performance of

actual pumps. A pump requires more work because of unavoidable losses due to friction and fluid turbulence. The work done by a pump W p is equal to the change in enthalpy across the actual pump.

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W p actual = (H out -H in ) actual (1-32) w ˙ p actual = m ˙ (h out -h in ) actual

(1-33) Pump efficiency, η p , is defined as the ratio of the work required by the pump if it were an ideal

pump w p, ideal to the actual work required by the pump w p, actual . W η p, ideal

(1-34)

W p, actual

Example:

A pump operating at 75% efficiency has an inlet specific enthalpy of 200 Btu/lbm. The exit specific enthalpy of the ideal pump is 600 Btu/lbm. What is the exit specific enthalpy of the actual pump?

Solution: Using Equation 1-34:

w p, ideal η p

w p, actual

w p, ideal

w p, actual

(h out h in ) ideal

(h out h in ) actual

(h out h in ) ideal

h out, actual

h in, actual

(600 Btu/lbm 200 Btu/lbm)

h out, actual

200 Btu/lbm

h out, actual = 533.3 Btu/lbm + 200 Btu/lbm

h out, actual = 733.3 Btu/lbm

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Pump efficiency, η p , relates the work required by an ideal pump to the actual work required by the pump; it relates the minimum amount of work theoretically possible to the actual work

required by the pump. However, the work required by a pump is normally only an intermediate form of energy. Normally a motor or turbine is used to run the pump. Pump efficiency does not account for losses in this motor or turbine. An additional efficiency factor, motor efficiency

η m , is defined as the ratio of the actual work required by the pump to the electrical energy input to the pump motor, when both are expressed in the same units.

W η p, actual

W m, in C

where: η m

= motor efficiency (no units) W p, actual

= actual work required by the pump (ft-lbf) W m, in

= electrical energy input to the pump motor (kw-hr)

C = conversion factor = 2.655 x 10 6 ft-lbf/kw-hr

Like pump efficiency η p , motor efficiency η m is always less than 1.0 or 100% for an actual pump motor. The combination of pump efficiency η p and motor efficiency η m relates the ideal pump to the electrical energy input to the pump motor.

W p, ideal

η m η p (1-35)

W m, in C

where: η m

= motor efficiency (no units)

= pump efficiency (no units)

W p, ideal = ideal work required by the pump (ft-lbf) W m, in

= electrical energy input to the pump motor (kw-hr)

C = conversion factor = 2.655 x 10 6 ft-lbf/kw-hr

A heat exchanger is designed to transfer heat between two working fluids. There are several heat exchangers used in power plant steam cycles. In the steam generator or boiler, the heat source (e.g., reactor coolant) is used to heat and vaporize the feedwater. In the condenser, the steam exhausting from the turbine is condensed before being returned to the steam generator. In addition to these two major heat exchangers, numerous smaller heat exchangers are used throughout the steam cycle. Two primary factors determine the rate of heat transfer and the temperature difference between the two fluids passing through the heat exchanger.

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In the application of the first law general energy equation to a simple heat exchanger under steady flow conditions, it is found that the mass flow rates and enthalpies of the two fluids are related by the following relationship.

m ˙ 1 (h out, 1 h in, 1 )

m ˙ 2 (h out, 2 h in, 2 )

(1-36)

where: m ˙ 1 = mass flow rate of the working fluid 1 (lbm/hr)

m ˙ 2 = mass flow rate of the working fluid 2 (lbm/hr)

h out, 1 = specific enthalpy of the working fluid 1 leaving the heat exchanger (Btu/lbm)

h in, 1 = specific enthalpy of the working fluid 1 entering the heat exchanger (Btu/lbm)

h out, 2 = specific enthalpy of the working fluid 2 leaving the heat exchanger (Btu/lbm)

h in, 2 = specific enthalpy of the working fluid 2 entering the heat exchanger (Btu/lbm)

In the preceding sections we have discussed the Carnot cycle, cycle efficiencies, and component efficiencies. In this section we will apply this information to allow us to compare and evaluate various ideal and real cycles. This will allow us to determine how modifying a cycle will affect the cycle’s available energy that can be extracted for work.

Since the efficiency of a Carnot cycle is solely dependent on the temperature of the heat source and the temperature of the heat sink, it follows that to improve a cycles’ efficiency all we have to do is increase the temperature of the heat source and decrease the temperature of the heat sink. In the real world the ability to do this is limited by the following constraints.

1. For a real cycle the heat sink is limited by the fact that the "earth" is our final heat sink. And therefore, is fixed at about 60°F (520°R).

2. The heat source is limited to the combustion temperatures of the fuel to be burned or the maximum limits placed on nuclear fuels by their structural components (pellets, cladding etc.). In the case of fossil fuel cycles the upper limit is ~3040°F (3500°R). But even this temperature is not attainable due to the metallurgical restraints of the boilers, and therefore they are limited to about 1500°F (1960°R) for a maximum heat source temperature.

Using these limits to calculate the maximum efficiency attainable by an ideal Carnot cycle gives the following.

1960 o R 520 η R

T SINK

SOURCE

T SOURCE

1960 o

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This calculation indicates that the Carnot cycle, operating with ideal components under real world constraints, should convert almost 3/4 of the input heat into work. But, as will be shown, this ideal efficiency is well beyond the present capabilities of any real systems.

Heat Rejection

To understand why an efficiency of 73% is not possible we must analyze the Carnot cycle, then compare the cycle using real and ideal components. We will do this by looking at the T-s diagrams of Carnot cycles using both real and ideal components.

The energy added to a working fluid during the Carnot isothermal expansion is given by q s . Not all of this energy is available for use by the heat engine since a portion of it (q r ) must be rejected to the environment. This is given by:

q r =T o ∆ s in units of Btu/lbm, (1-37) where T o is the average heat sink temperature of 520°R. The available energy (A.E.) for the

Carnot cycle may be given as: s A.E. = q -q r .

(1-38) Substituting equation 1-37 for q r gives:

s A.E. = q -T o ∆ s in units of Btu/lbm. (1-39) and is equal to the area of the shaded

region labeled available energy in Figure 28 between the temperatures 1962° and 520°R. From Figure 28 it can been seen that any cycle operating at a temperature of less than 1962°R will be less efficient. Note that by developing materials capable of withstanding the stresses above 1962°R, we could greatly add to the energy available for use by the plant cycle.

From equation 1-37, one can see why the change in entropy can be defined as a measure of the energy unavailable to do work. If the temperature of the heat sink is known, then the change in entropy does correspond to a measure

Figure 28 Carnot Cycle

of the heat rejected by the engine.

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Figure 29 Carnot Cycle vs. Typical Power Cycle Available Energy

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Figure 29 is a typical power cycle employed by a fossil fuel plant. The working fluid is water, which places certain restrictions on the cycle. If we wish to limit ourselves to operation at or below 2000 psia, it is readily apparent that constant heat addition at our maximum temperature of 1962°R is not possible (Figure 29, 2’ to 4). In reality, the nature of water and certain elements of the process controls require us to add heat in a constant pressure process instead (Figure 29, 1-2-3-4). Because of this, the average temperature at which we are adding heat is far below the maximum allowable material temperature.

As can be seen, the actual available energy (area under the 1-2-3-4 curve, Figure 29) is less than half of what is available from the ideal Carnot cycle (area under 1-2’-4 curve, Figure 29) operating between the same two temperatures. Typical thermal efficiencies for fossil plants are on the order of 40% while nuclear plants have efficiencies of the order of 31%. Note that these numbers are less than 1/2 of the maximum thermal efficiency of the ideal Carnot cycle calculated earlier.

Figure 30 shows a proposed Carnot steam cycle superimposed on a T-s diagram. As shown, it has several problems which make it undesirable as a practical power cycle. First a great deal of pump work is required to compress a two phase mixture of water and steam from point 1 to the saturated liquid state at point 2. Second, this same isentropic compression will probably result in some pump cavitation in the feed system. Finally, a condenser designed to produce a two- phase mixture at the outlet (point 1) would pose technical problems.

Figure 30 Ideal Carnot Cycle

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Early thermodynamic developments were centered around improving the performance of contemporary steam engines. It was desirable to construct a cycle that was as close to being reversible as possible and would better lend itself to the characteristics of steam and process control than the Carnot cycle did. Towards this end, the Rankine cycle was developed.

The main feature of the Rankine cycle, shown in Figure 31, is that it confines the isentropic compression process to the liquid phase only (Figure 31 points 1 to 2). This minimizes the amount of work required to attain operating pressures and avoids the mechanical problems associated with pumping a two-phase mixture. The compression process shown in figure 31

between points 1 and 2 is greatly exaggerated * . In reality, a temperature rise of only 1°F occurs in compressing water from 14.7 psig at a saturation temperature of 212°F to 1000 psig.

Figure 31 Rankine Cycle

* The constant pressure lines converge rapidly in the subcooled or compressed liquid region and it is difficult to distinguish them from the saturated liquid line without artificially expanding them away from it.

In a Rankine cycle available and unavailable energy on a T-s diagram, like a T-s diagram of a Carnot cycle, is represented by the areas under the curves. The larger the unavailable energy, the less efficient the cycle.

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Figure 32 Rankine Cycle With Real v.s. Ideal

From the T-s diagram (Figure 32) it can also be seen that if an ideal component, in this case the turbine, is replaced with a non-ideal component, the efficiency of the cycle will be reduced. This is due to the fact that the non-ideal turbine incurs an increase in entropy which increases the area under the T-s curve for the cycle. But the increase in the area of available energy (3-2-3’, Figure 32) is less than the increase in area for unavailable energy (a-3-3’-b, Figure 32).

Figure 33 Rankine Cycle Efficiencies T-s

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The same loss of cycle efficiency can be seen when two Rankine cycles are compared (see Figure 33). Using this type of comparison, the amount of rejected energy to available energy of one cycle can be compared to another cycle to determine which cycle is the most efficient, i.e. has the least amount of unavailable energy.

An h-s diagram can also be used to compare systems and help determine their efficiencies.

Like the T-s diagram, the h-s diagram will show (Figure 34) that substituting non-ideal components in place of ideal components in a cycle, will result in the reduction in the cycles efficiency.

This is

because a change in enthalpy (h) always occurs when work is done or heat is added or removed in an actual cycle (non-ideal). This deviation

Figure 34 h-s Diagram

from an ideal constant enthalpy (vertical line on the diagram) allows the inefficiencies of the cycle to be easily seen on a h-s diagram.

Typical Steam Cycle

Figure 35 shows a simplified version of the major components of a typical steam plant cycle. This is a simplified version and does not contain the exact detail that may be found at most power plants. However, for the purpose of understanding the basic operation of a power cycle, further detail is not necessary.

The following are the processes that comprise the cycle: 1-2: Saturated steam from the steam generator is expanded in the high pressure (HP)

turbine to provide shaft work output at a constant entropy. 2-3: The moist steam from the exit of the HP turbine is dried and superheated in the

moisture separator reheater (MSR). 3-4: Superheated steam from the MSR is expanded in the low pressure (LP) turbine to

provide shaft work output at a constant entropy.

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4-5: Steam exhaust from the turbine is condensed in the condenser in which heat is

transferred to the cooling water under a constant vacuum condition. 5-6: The feedwater is compressed as a liquid by the condensate and feedwater pump

and the feedwater is preheated by the feedwater heaters. 6-1: Heat is added to the working fluid in the steam generator under a constant

pressure condition.

Figure 35 Typical Steam Cycle

The previous cycle can also be represented on a T-s diagram as was done with the ideal Carnot and Rankine cycles. This is shown in Figure 36. The numbered points on the cycle correspond to the numbered points on Figure 36.

It must be pointed out that the cycle we have just shown is an ideal cycle and does not exactly represent the actual processes in the plant. The turbine and pumps in an ideal cycle are ideal pumps and turbines and therefore do not exhibit an increase in entropy across them. Real pumps and turbines would exhibit an entropy increase across them.

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Figure 36 Steam Cycle (Ideal)

Figure 37 is a T-s diagram of a cycle which more closely approximates actual plant processes. The pumps and turbines in this cycle more closely approximate real pumps and turbines and thus exhibit an entropy increase across them. Additionally, in this cycle, a small degree of subcooling is evident in the condenser as shown by the small dip down to point 5. This small amount of subcooling will decrease cycle efficiency since additional heat has been removed from the cycle to the cooling water as heat rejected. This additional heat rejected must then be made up for in the steam generator. Therefore, it can be seen that excessive condenser subcooling will decrease cycle efficiency. By controlling the temperature or flow rate of the cooling water to the condenser, the operator can directly effect the overall cycle efficiency.

Figure 37 Steam Cycle (Real)

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It is sometimes useful to plot on the Mollier diagram the processes that occur during the cycle. This is done on Figure 38. The numbered points on Figure 38 correspond to the numbered points on Figures 35 and 36. Because the Mollier diagram is a plot of the conditions existing for water in vapor form, the portions of the plot which fall into the region of liquid water do not show up on the Mollier diagram. The following conditions were used in plotting the curves on Figure 38.

Point 1: o Saturated steam at 540 F Point 2:

82.5% quality at exit of HP turbine Point 3: o Temperature of superheated steam is 440 F

Point 4: Condenser vacuum is 1 psia The solid lines on Figure 38 represent the conditions for a cycle which uses ideal turbines as

verified by the fact that no entropy change is shown across the turbines. The dotted lines on Figure 38 represent the path taken if real turbines were considered, in which case an increase in entropy is evident.

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Causes of Inefficiency

In the preceeding sections, cycle and component efficiencies have been discussed, but the actual causes or reasons for the inefficiencies have not been explained. In this section we will compare some of the types and causes for the inefficiencies of real components and cycles to that of their "ideal" counterparts.

Components

In real systems, a percentage of the overall cycle inefficiency is due to the losses by the individual components. Turbines, pumps, and compressors all behave non-ideally due to heat losses, friction and windage losses. All of these losses contribute to the non- isentropic behavior of real equipment. As explained previously (Figures 24, 25) these losses can be seen as an increase in the system’s entropy or amount of energy that is unavailable for use by the cycle.

Cycles

In real systems, a second source of inefficiencies is from the compromises made due to cost and other factors in the design and operation of the cycle. Examples of these types of losses are: In a large power generating station the condensers are designed to subcool the liquid by 8-10°F. This subcooling allows the condensate pumps to pump the water forward without cavitation. But, each degree of subcooling is energy that must be put back by reheating the water, and this heat (energy) does no useful work and therefore increases the inefficiency of the cycle. Another example of a loss due to a system’s design is heat loss to the environment, i.e. thin or poor insulation. Again this is energy lost to the system and therefore unavailable to do work. Friction is another real world loss, both resistance to fluid flow and mechanical friction in machines. All of these contribute to the system’s inefficiency.

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Summary

The important information from this chapter is summarized below.

Second Law of Thermodynamics Summary

• Planck’s statement of the Second Law of Thermodynamics is:

It is impossible to construct an engine that will work in a complete cycle and produce no other effect except the raising of

a weight and the cooling of a heat reservoir. •

The Second Law of Thermodynamics demonstrates that the maximum possible efficiency of a system is the Carnot efficiency written as:

η = (T H -T C )/T H

• The maximum efficiency of a closed cycle can be determined by calculating the efficiency of a Carnot cycle operating between the same value of high and low temperatures.

• The efficiency of a component can be calculated by comparing the work produced by the component to the work that would have been produced by an ideal component operating isentropically between the same inlet and outlet conditions.

• An isentropic expansion or compression process will be represented as a vertical line on a T-s or h-s diagram. A real expansion or compression process will look similar, but will be slanted slightly to the right.

• Efficiency will be decreased by: Presence of friction

Heat losses Cycle inefficiencies

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